Compact orthoalgebras

Author:
Alexander Wilce

Journal:
Proc. Amer. Math. Soc. **133** (2005), 2911-2920

MSC (2000):
Primary 06F15, 06F30; Secondary 03G12, 81P10

DOI:
https://doi.org/10.1090/S0002-9939-05-07884-6

Published electronically:
May 2, 2005

MathSciNet review:
2159769

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Abstract | References | Similar Articles | Additional Information

Abstract: We initiate a study of topological orthoalgebras (TOAs), concentrating on the compact case. Examples of TOAs include topological orthomodular lattices, and also the projection lattice of a Hilbert space. As the latter example illustrates, a lattice-ordered TOA need not be a topological lattice. However, we show that a compact Boolean TOA is a topological Boolean algebra. Using this, we prove that any compact regular TOA is atomistic , and has a compact center. We prove also that any compact TOA with isolated is of finite height. We then focus on *stably ordered* TOAs: those in which the upper set generated by an open set is open. These include both topological orthomodular lattices and interval orthoalgebras - in particular, projection lattices. We show that the topology of a compact stably-ordered TOA with isolated is determined by that of its space of atoms.

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Additional Information

**Alexander Wilce**

Affiliation:
Department of Mathematics, Susquehanna University, Selinsgrove, Pennsylvania 17870

Email:
wilce@susqu.edu

DOI:
https://doi.org/10.1090/S0002-9939-05-07884-6

Keywords:
Orthoalgebra,
effect algebra,
orthomodular lattice,
topological lattice,
quantum logic

Received by editor(s):
August 22, 2003

Received by editor(s) in revised form:
June 10, 2004

Published electronically:
May 2, 2005

Communicated by:
Lance W. Small

Article copyright:
© Copyright 2005
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.